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A new mm algorithm for root-finding problems

STATISTICA NEERLANDICA 2024年第4期78卷 692-701页

作者： Li, Xun-Jian Li, Shuang Tian, Guo-Liang Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China Southern Univ Sci & Technol Dept Math Shenzhen Peoples R China Southern Univ Sci & Technol Dept Stat & Data Sci Shenzhen 518055 Guangdong Peoples R China

The minorization-maximization (mm) algorithm is an optimization technique for iteratively calculating the maximizer of a concave target function rather than a root-finding tool. In this paper, we in the first time develop the mm algorithm as a new method for seeking the root x* of a univariate nonlinear equation g(x) = 0. The key idea is to transfer the root-finding issue to iteratively calculate the maximizer of a concave target function by designing a new mm algorithm. According to the ascent property of the mm algorithm, we know that the proposed algorithm converges to the root x* and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm.

关键词： integral version of Taylor expansion mm algorithm Newton's method optimization univariate nonlinear equation

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An mm algorithm to Estimate Parameters in Continuous-Time Markov Chains 1

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20th International Conference on Quantitative Evaluation of Systems (QEST)

作者： Bacci, Giovanni Ingolfsdottir, Anna Larsen, Kim G. Reynouard, Raphael Aalborg Univ Dept Comp Sci Aalborg Denmark Reykjavik Univ Dept Comp Sci Reykjavik Iceland

ISBN: (数字)9783031438356

ISBN: (纸本)9783031438349;9783031438356

Prism and Storm are popular model checking tools that provide a number of powerful analysis techniques for Continuous-time Markov chains (CTMCs). The outcome of the analysis is strongly dependent on the parameter values used in the model which govern the timing and probability of events of the resulting CTMC. However, for some applications, parameter values have to be empirically estimated from partially-observable executions. In this work, we address the problem of estimating parameter values of CTMCs expressed as Prism models from a number of partially-observable executions whichmight possiblymiss some dwell time measurements. The semantics of the model is expressed as a parametricCTMC(pCTMC), i.e., CTMC where transition rates are polynomial functions over a set of parameters. Then, building on a theory of algorithms known by the initials mm, for minorization-maximization, we present an iterative maximum likelihood estimation algorithm for pCTMCs. We present an experimental evaluation of the proposed technique on a number of CTMCs from the quantitative verification benchmark set. We conclude by illustrating the use of our technique in a case study: the analysis of the spread of COVID-19 in presence of lockdown countermeasures.

关键词： mm algorithm Continuous-time Markov chains Maximum likelihood estimation

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An mm algorithm for estimation of a two component semiparametric density mixture with a known component

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ELECTRONIC JOURNAL OF STATISTICS 2018年第1期12卷 1181-1209页

作者： Shen, Zhou Levine, Michael Shang, Zuofeng Purdue Univ Dept Stat 250 N Univ St W Lafayette IN 47907 USA IUPUI Dept Math Sci 402 N Blackford St Indianapolis IN 46202 USA

We consider a semiparametric mixture of two univariate density functions where one of them is known while the weight and the other function are unknown. We do not assume any additional structure on the unknown density function. For this mixture model, we derive a new sufficient identifiability condition and pinpoint a specific class of distributions describing the unknown component for which this condition is mostly satisfied. We also suggest a novel approach to estimation of this model that is based on an idea of applying a maximum smoothed likelihood to what would otherwise have been an ill-posed problem. We introduce an iterative mm (Majorization-Minimization) algorithm that estimates all of the model parameters. We establish that the algorithm possesses a descent property with respect to a log-likelihood objective functional and prove that the algorithm, indeed, converges. Finally, we also illustrate the performance of our algorithm in a simulation study and apply it to a real dataset.

关键词： Penalized smoothed likelihood mm algorithm regularization

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Quasi-Newton Acceleration of EM and mm algorithms via Broyden's Method

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2024年第2期33卷 393-406页

作者： Agarwal, Medha Xu, Jason Univ Washington Dept Stat Seattle WA USA Duke Univ Dept Stat Sci Durham NC 27708 USA

The principle of majorization-minimization (mm) provides a general framework for eliciting effective algorithms to solve optimization problems. However, the resulting methods often suffer from slow convergence, especially in large-scale and high-dimensional data settings. This has motivated several acceleration schemes tailored for mm algorithms, but many existing approaches are either problem-specific, or rely on approximations and heuristics loosely inspired by the optimization literature. We propose a novel quasi-Newton method for accelerating any valid mm algorithm, cast as seeking a fixed point of the mm algorithm map. The method does not require specific information or computation from the objective function or its gradient, and enjoys a limited-memory variant amenable to efficient computation in high-dimensional settings. By rigorously connecting our approach to Broyden's classical root-finding methods, we establish convergence guarantees and identify conditions for linear and super-linear convergence. These results are validated numerically and compared to peer methods in a thorough empirical study, showing that it achieves state-of-the-art performance across a diverse range of problems. Supplementary materials for this article are available online.

关键词： Broyden's root finding method mm algorithm quasi-Newton method

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A new mm algorithm for constrained estimation in the proportional hazards model

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2015年 84卷 135-151页

作者： Ding, Jieli Tian, Guo-Liang Yuen, Kam Chuen Wuhan Univ Sch Math & Stat Wuhan 430072 Hubei Peoples R China Univ Hong Kong Dept Stat & Actuarial Sci Hong Kong Hong Kong Peoples R China

The constrained estimation in Cox's model for the right-censored survival data is studied and the asymptotic properties of the constrained estimators are derived by using the Lagrangian method based on Karush-Kuhn-Tucker conditions. A novel minorization-maximization (mm) algorithm is developed for calculating the maximum likelihood estimates of the regression coefficients subject to box or linear inequality restrictions in the proportional hazards model. The first M-step of the proposed mm algorithm is to construct a surrogate function with a diagonal Hessian matrix, which can be reached by utilizing the convexity of the exponential function and the negative logarithm function. The second M-step is to maximize the surrogate function with a diagonal Hessian matrix subject to box constraints, which is equivalent to separately maximizing several one-dimensional concave functions with a lower bound and an upper bound constraint, resulting in an explicit solution via a median function. The ascent property of the proposed mm algorithm under constraints is theoretically justified. Standard error estimation is also presented via a non-parametric bootstrap approach. Simulation studies are performed to compare the estimations with and without constraints. Two real data sets are used to illustrate the proposed methods. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Asymptotic properties Bootstrap approach Constrained estimation Karush-Kuhn-Tucker conditions mm algorithm Proportional hazards model

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Accelerated mm algorithms for Inference of Ranking Scores from Comparison Data

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OPERATIONS RESEARCH 2023年第4期71卷 1318-1342页

作者： Vojnovic, Milan Yun, Se-Young Zhou, Kaifang London Sch Econ & Polit Sci Dept Stat London England Korea Adv Inst Sci & Technol Kim Jaechul Grad Sch Artificial Intelligence Daejeon 34141 South Korea

The problem of assigning ranking scores to items based on observed comparison data (e.g., paired comparisons, choice, and full ranking outcomes) has been of continued interest in a wide range of applications, including information search, aggregation of social opinions, electronic commerce, online gaming platforms, and, more recently, evaluation of machine learning algorithms. The key problem is to compute ranking scores, which are of interest for quantifying the strength of skills, relevancies, or preferences, and the prediction of ranking outcomes when the ranking scores are estimates of parameters of a statistical model of ranking outcomes. One of the most popular statistical models of ranking outcomes is the Bradley-Terry model for paired comparisons (equivalent to multinomial logit model) and its extensions to choice and full ranking outcomes. The problem of computing ranking scores under these statistical models amounts to estimating model parameters. In this paper, we study a popular method for inference of the Bradley-Terry model parameters-namely, the mm algorithm, where mm refers to minorization-maximization-for maximum likelihood estimation and maximum a posteriori probability estimation. This class of models includes the Bradley-Terry model of paired comparisons, the Rao-Kupper model of paired comparisons allowing for tie outcomes, the Luce choice model, and the Plackett-Luce ranking model. We establish tight characterizations of the convergence rate for the mm algorithm and show that it is essentially equivalent to that of a gradient descent algorithm. For the maximum likelihood estimation, the convergence is shown to be linear with the rate crucially determined by the algebraic connectivity of the matrix of item pair co-occurrences in observed comparison data. For the Bayesian inference, the convergence rate is also shown to be linear, with the rate determined by a parameter of the prior distribution in a way that can make the convergence arbitrarily s

关键词： rank aggregation generalized Bradley-Terry model maximum likelihood estimation Bayesian inference convex optimization gradient descent mm algorithm rate of convergence

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Wilcoxon-type multivariate cluster elastic net

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NEUROCOMPUTING 2025年 624卷

作者： Hiraishi, Mayu Tanioka, Kensuke Yadohisa, Hiroshi Doshisha Univ Grad Sch Culture & Informat Sci Kyotanabe Kyoto 6100394 Japan Doshisha Univ Dept Biomed Sci & Informat Kyotanabe Kyoto 6100394 Japan Doshisha Univ Dept Culture & Informat Sci Kyotanabe Kyoto 6100394 Japan

We propose a method for high dimensional multivariate regression that is robust to heavy-tailed distributions or outliers, while preserving estimation accuracy in normal random error distributions. We extend the Wilcoxontype regression to a multivariate regression model as a tuning-free approach to robustness. Furthermore, the proposed method regularizes the L1 and L2 terms of the clustering based on k-means, which is extended from the multivariate cluster elastic net. The estimation of the regression coefficient and variable selection are produced simultaneously. Moreover, considering the relationship among the correlation of response variables through the clustering is expected to improve the estimation performance. The numerical simulation demonstrates that our proposed method overperforms the multivariate cluster method and other multiple regression methods in the case of heavy-tailed error distribution and outliers. The proposed method also indicates stability in normal error distribution. Finally, we confirm the efficacy of our proposed method using gene data.

关键词： mm algorithm Multivariate regression Robust statistics

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A Majorization-Minimization algorithm for Neuroimage Registration

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SIAM JOURNAL ON IMAGING SCIENCES 2024年第1期17卷 273-300页

作者： Zhou, Gaiting Tward, Daniel Lange, Kenneth UCLA Computat Med Los Angeles CA 90024 USA

Intensity -based image registration is critical for neuroimaging tasks, such as 3D reconstruction, times -series alignment, and common coordinate mapping. The gradient -based optimization methods commonly used to solve this problem require a careful selection of step -length. This limitation imposes substantial time and computational costs. Here we propose a gradient -independent rigid -motion registration algorithm based on the majorization-minimization (mm) principle. Each iteration of our intensity -based mm algorithm reduces to a simple point -set rigid registration problem with a closed form solution that avoids the step -length issue altogether. The details of the algorithm are presented, and an error bound for its more practical truncated form is derived. The performance of the mm algorithm is shown to be more effective than gradient descent on simulated images and Nissl stained coronal slices of mouse brain. We also compare and contrast the similarities and differences between the mm algorithm and another gradient -free registration algorithm called the block -matching method. Finally, extensions of this algorithm to more complex problems are discussed.

关键词： mm algorithm image registration neuroimaging

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Parameter-expanded ECME algorithms for logistic and penalized logistic regression

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COMPUTATIONAL STATISTICS 2025年 1-27页

作者： Henderson, Nicholas C. Ouyang, Zhongzhe Univ Michigan Dept Biostat 1415 Washington Hts Ann Arbor MI 48109 USA

Parameter estimation in logistic regression is a well-studied problem with the Newton-Raphson method being one of the most prominent optimization techniques used in practice. A number of monotone optimization methods including minorization-maximization (mm) algorithms, expectation-maximization (EM) algorithms and related variational Bayes approaches offer useful alternatives guaranteed to increase the logistic regression likelihood at every iteration. In this article, we propose and evaluate an optimization procedure that is based on a straightforward modification of an EM algorithm for logistic regression. Our method can substantially improve the computational efficiency of the EM algorithm while preserving the monotonicity of EM and the simplicity of the EM parameter updates. By introducing an additional latent parameter and selecting this parameter to maximize the penalized observed-data log-likelihood at every iteration, our iterative algorithm can be interpreted as a parameter-expanded expectation-conditional maximization either (ECME) algorithm, and we demonstrate how to use the parameter-expanded ECME with an arbitrary choice of weights and penalty function. In addition, we describe a generalized version of our parameter-expanded ECME algorithm that can be tailored to the challenges encountered in specific high-dimensional problems, and we study several interesting connections between this generalized algorithm and other well-known methods. Performance comparisons between our method, the EM algorithm, Newton-Raphson, and several other optimization methods are presented using an extensive series of simulation studies based upon both real and synthetic datasets.

关键词： Convergence acceleration EM algorithm mm algorithm P & oacute lya-Gamma augmentation Parameter expansion

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A unified Minorization-Maximization approach for estimation of general mixture models

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Applied Mathematics(A Journal of Chinese Universities) 2024年第2期39卷 343-362页

作者： HUANG Xi-fen LIU Deng-ge ZHOU Yun-peng ZHU Fei School of Mathematics Yunnan Normal UniversityKunming 650092China Department of Statistics and Actuarial Science The University of Hong KongHong KongChina

The mixed distribution model is often used to extract information from heteroge-neous data and perform modeling *** the density function of mixed distribution is complicated or the variable dimension is high,it usually brings challenges to the parameter es-timation of the mixed distribution *** application of mm algorithm can avoid complex expectation calculations,and can also solve the problem of high-dimensional optimization by decomposing the objective *** this paper,mm algorithm is applied to the parameter estimation problem of mixed distribution *** method of assembly and decomposition is used to construct the substitute function with separable parameters,which avoids the problems of complex expectation calculations and the inversion of high-dimensional matrices.

关键词： mm algorithm mixed distribution model parameter estimation assembly decomposition tech-nology parameter separation

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